From what I recall, the game doesn't outright give a reason for certain things, most likely so players can draw their own thoughts and conclusions themselves. (Or at least, Markus seems to be in favor of that, and I'd consider him the 'True Route' of the game since his special ending has the most requirements.)
There are a few facts we can take from the game itself, though, so I can try to answer your questions from what I've played of the game (along with my own speculation).
!Spoilers and Speculation below!
1. I assume that Lazarus appeared in the dream because Markus 'pulled' him there. Andrei is able to appear in MC's dreams to introduce himself, so it's possible that it's a Golgotha ability that allows them to visit and invite others to a dream.
2. The connection between MC and Lazarus could be several things. On the day Lazarus was to be turned into a vampire, both of them were at the club when MC was turned instead. The two of them share a connection to Markus, who introduced both of them to vampires and their world in different aspects. They share a connection with Saorise, who supported both of them and aims to have them join her faction. They even share a slight connection with Andrei, as Lazarus and MC caught his attention (albeit, at different points in time, and for wildly different reasons). Even the way MC and Lazarus meet is by pure coincidence, with Saorise assigning MC to a murder case that leads them to Lazarus.
Honestly, there are so many coincidences between MC and Lazarus, the people they meet, how those people influence them, and how they just happen to be in certain places at the same time (or connect them to one another) that it's bizarre. I suppose you could say their fates are intertwined?
The connection could also be much simpler, as Lazarus feels intense envy and hatred towards MC for taking his place as Saorise's vampire, so he may be connected to the MC (and been able to enter the dream, with help from Markus' Golgotha power) purely out of spite.
3. I think it's just a personal (or RP) choice? I haven't tried a playthrough with Markus as my mentor where I agree to Lazarus' deal and save him, but I imagine the stress of confronting Markus and everything that followed may have taken the fight out of him, at least emotionally.
4. Markus is a troll, and the MC's Sire is leader of House Troll.
5. Markus does love MC. It's somewhat implied that he feels great affection for MC regardless of whether you enter into a romance with him or not, but I wouldn't call his love familial. The reason he's relieved about not having sex with MC is because he's Asexual and, judging by a few lines, sex-repulsed.
If you chose the 'Does this mean our relationship is a friendship?' option, Markus states that he likes affection and feelings that extend beyond a 'normal' friendship, he just simply doesn't care for sex in a relationship, since the act itself makes him feel deeply uncomfortable and causes him to dissociate.
As for Markus' love itself, he confesses that while it may not follow the usual 'love' MC may be acquainted with, he doesn't apply any labels to it. Terms, yes, but he never says that it's specifically platonic, familial, or otherwise. Just that, for him, there is only one type of love , and that he feels it greatest for MC (at a distance).
As for his relationship with Aleusha, while she does describe it as familial, she also makes it clear that there isn't any intimacy or romantic feelings between them - on her part or his. Markus is fond of and respects her, yes, but there are distinctions in his relationship with her and the MC.
TL;DR: Yes, Markus does love MC. He has no interest in sex as, regardless of who his partner is, it feels alien and wrong to him. While he may not want to have sex with MC, he expresses that he loves them (in his own way), likes affection and non-sexual intimacy, and that their relationship doesn't have to be a friendship, unless that's what MC wants. His relationship with Aleusha is different compared to his relationship with MC, but it's doubtful that it has any of the undertones of intimacy that Markus has with MC.
!Spoilers and Speculation over!
As for a guide, there is a walkthrough over on the Steam community page that Dugodder mentioned above. Don't worry about your English - it's very good, and perfectly understandable!